Reverse-BSDE Monte Carlo

TL;DR

This work reframes diffusion-based generative modeling as a sampling problem by formulating it as a Forward-Backward SDE (FBSDE), allowing direct sampling from densities known up to a normalization constant $p_0(\boldsymbol{x})=\pi(\boldsymbol{x})/c$ without pre-estimating the score $\nabla \log p$. It introduces a Deep BSDE framework that parameterizes the forward and backward components with neural networks and uses Euler–Maruyama discretization to solve the coupled FBSDE, ensuring $X_T \sim p_0$ and enabling sampling in high dimensions. Theoretical contributions include a uniqueness result for the FBSDE under standard conditions and a convergence-type bound for the discrete-time Deep BSDE scheme, justifying the proposed loss and training procedure. Numerically, the method demonstrates capability to sample from distributions up to a normalization constant, as shown on a 1D Gaussian and a 2D Gaussian mixture, highlighting practical potential for Bayesian sampling tasks and broader diffusion-model applications. Overall, the paper connects diffusion-based generative modeling with stochastic control and PDE-based sampling, offering a score-free, neural-network-based approach to draw samples from complex, high-dimensional distributions where the normalization constant is unknown.

Abstract

Recently, there has been a growing interest in generative models based on diffusions driven by the empirical robustness of these methods in generating high-dimensional photorealistic images and the possibility of using the vast existing toolbox of stochastic differential equations. %This remarkable ability may stem from their capacity to model and generate multimodal distributions. In this work, we offer a novel perspective on the approach introduced in Song et al. (2021), shifting the focus from a "learning" problem to a "sampling" problem. To achieve this, we reformulate the equations governing diffusion-based generative models as a Forward-Backward Stochastic Differential Equation (FBSDE), which avoids the well-known issue of pre-estimating the gradient of the log target density. The solution of this FBSDE is proved to be unique using non-standard techniques. Additionally, we propose a numerical solution to this problem, leveraging on Deep Learning techniques. This reformulation opens new pathways for sampling multidimensional distributions with densities known up to a normalization constant, a problem frequently encountered in Bayesian statistics.

Figures (5)

• Figure 1: Sample of Normal(2, 0.3) generated by the proposed algorithm (in blue). The red line is the exact density.
• Figure 2: Sample of a mixture of nine bivariate normal distributions generated by the proposed algorithm.
• Figure 3: Difference between estimated and real scores in the Gaussian example for $t=0,1,2$ and 3.
• Figure 4: Difference of the between the first dimension of estimated and real scores of the Mixture of Gaussians example for $t=0,1,2$ and 3.
• Figure 5: Difference of the between the second dimension of estimated and real scores of the Mixture of Gaussians example for $t=0,1,2$ and 3.

Theorems & Definitions (6)

• Remark 1
• Lemma 1
• Theorem 1
• Theorem 2: from han2020convergence
• proof : Proof of Lemma \ref{['lemma:reverse']}
• proof